This work concerns the representation theory and cohomology of a finite unipotent supergroup scheme G over a perfect field k of positive characteristic \(p\ge 3\). It is proved that an element x in the cohomology of G is nilpotent if and only if for every extension field K of k and every elementary sub-supergroup scheme \(E\subseteq G_K\), the restriction of \(x_K\) to E is nilpotent. It is also shown that a kG-module M is projective if and only if for every extension field K of k and every elementary sub-supergroup scheme \(E\subseteq G_K\), the restriction of \(M_K\) to E is projective. The statements are motivated by, and are analogues of, similar results for finite groups and finite group schemes, but the structure of elementary supergroups schemes necessary for detection is more complicated than in either of these cases. One application is a detection theorem for the nilpotence of cohomology, and projectivity of modules, over finite dimensional Hopf subalgebras of the Steenrod algebra.

这项工作涉及在正特性\（p \ ge 3 \）的理想场k上的有限单能超群方案G的表示理论和同调性。证明了一个元素X在的同调ģ是幂零当且仅当对于每个扩展字段ķ的ķ和每初级子超组方案\（E \ subseteq G_K \） ，所述的限制\（X_K \）到E是无能的。还表明，当且仅当对于k的每个扩展字段K，kG模块M都是射影和每个基本的子超群方案\（E \ subseteq G_K \），\（M_K \）对E的限制是射影。这些陈述受有限组和有限组方案的相似结果的推动，并且是类似结果，但是检测所必需的基本超组方案的结构比这两种情况中的任何一种都更为复杂。一种应用是在Steenrod代数的有限维Hopf子代数上的同调性和模块的投影性的幂等性的检测定理。